Recommended: Find time to watch this talk by Seymour Papert.
Author: David Wees (page 24 of 97)
There are some ideas in mathematics which are powerful tools for thinking. I’ve enumerated some of them below.
- Equivalence
This is the idea that some things are the same as each other, and that some things are not. We see equivalence taught through early arithmetic and algebra. We also see this idea taught through trigonometric identities and proofs.
I watched a colleague of mine working with his class to introduce trigonometric identities (such as sin2x + cos2x = 1) and over and over again the idea of equivalence between different expressions came up. At one point a student tried to cancel the 1s in the following expression: $\frac{cos^2x-1}{cosx+1}$. My colleague then asked students to look at $\frac{2+1}{3+1}$ and students noticed that $\frac{3}{4}$ is not the same as $\frac{2}{3}$ which means that it must not be possible to do the same thing with the trigonometric expression. This idea is only possible to understand if you understand the bigger idea of equivalence.
- Sequencing
We see this idea comes up in patterns, ordering numbers, but it also comes up in limits. In the earliest years, the idea of sequencing is introduced as children learn to count by ones, twos, and so on, and in the later years, students look at lists of numbers and try and find a pattern to predict further numbers in the sequence.
In the final years of high school, if they learn calculus, students learn about how sequences apply to understanding the limit of a series. For example with $S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + … $, what should the value of S be to make this a true statement, assuming that the sequence of numbers continues forever? How does this idea relate to the idea of differentials and integrals?
More generally, any time we arrange anything in any kind of order, we necessarily have to use our ability to sequence ideas.
- Representation
This idea is where much of mathematical notation comes from as any mathematical notation is in fact a representation of another concept. Even number symbols are in fact representations of the idea of a number, rather than actually being the number themselves, eg. the numeral 5 is a representation of the idea of fiveness.
Another place representation comes up in a standard curriculum is when students explore functions, and they learn that functions can be graphed, given as ordered pairs, listed in a table, and represented by an equation. Each of these representations means the same thing, but written in a different form for our convenience.
- Precision
Early in school students learn the difference between an estimate and a calculation, and they learn that the way we use words (especially in mathematics) can sometimes lead to some interesting issues. For example, a common definition of a prime number is that it is any number which is divisible by one and itself. Unfortunately, many children read this definition and think that one is a prime number. Understanding that one has been chosen not to be included in the definition of the prime numbers so that we can more easily identify and use the properties of primes later on is something that many children either do not understand, or are never told about.
- Procedures
This is one powerful idea that most children spend a lot of time learning about. They learn that someone can create a procedure, or a list of steps to follow in order to achieve a particular goal, and that when that procedure is followed accurately, the result is predictable. This idea itself is seen in almost all areas of mathematics, but notably, it often fails when applied to mathematical problem solving because so many rich mathematical problems defy a simple procedure to solve.
- Subroutines
Learning how to decompose and reduce the complexity of problems with which we are faced is one of the most critical thinking tools we can learn. Over and over again, this issue comes up in mathematics. In fact, in order to support students ability to find and create subroutines is one of the reasons mathematics curriculums often follow similar sequences.
Here’s an activity: find a problem from secondary school mathematics, and work out how many different subroutines are necessary in order to solve that problem. One benefit of seeing this issue of subroutines is that it gives you a lot more empathy about what children have to go through in order to learn mathematics.
- Classification
This is the ability to take a list of traits, identify them in an object, and use those traits to understand the differences between one object and another. For example, if you look at the properties of a rectangle, you may recognize that opposite sides of the rectangle should be the same length, and that every angle in the rectangle should be $\frac{1}{4}$ of a full turn. One thing that many people struggle with is understanding that this definition of a rectangle also necessarily includes a sequence, and that in fact a square is a specific example of a form of rectangle.
Classification is one of the earliest thinking tools we learn. It is likely in fact that our ability to classify objects may be preprogrammed into our brains, given how early in life you can see children using it.
- Relationships
Mathematics is also about finding relationships between different objects. Look at the video below.
Is it surprising that through paper folding with a point and a circle that we can create the shape of a hyperbola?
Finding relationships between objects and finding how one area of mathematics is connected to another area of mathematics are critical skills to learn when solving mathematical problems. Some of the most challenging problems in mathematics have been solved by looking for parallels in other areas of mathematics that are easier to work with.
These powerful ideas also apply to non-mathematical problems. If students can learn these thinking skills, and learn how to apply them in a variety of different contexts, then they will hopefully be able to apply them to the problems they will face in life. Learning these skills empowers childrens to think.
What other powerful ideas do you think children should learn?
* Note that some of the examples on this page are best seen in the web version, rather than in email, or in your RSS reader.
I’m working on a presentation (for a staff meeting) on how to learn about technology. Technically, my title should be, "How to learn how to use technologies tools for specific tasks" but this isn’t very catchy.
I’d love to some feedback. Are there any techniques you use to learn how to use technology that I’ve not included?
Back in November I talked to my head of school to let him know that I was seeking leadership opportunities. I love my school, it is a fabulous place to work, and one of the things I love about it the most is that everyone has some shared responsibility for leadership in different areas of the school. We have two directors, one for the Junior School division, and one for the Senior School division, both of whom I admire and respect very much, and we have many people with minor leadership roles, essentially creating a somewhat flat hierarchy. One of the problems with this hierarchy though is that I have no way to gain enough experiences at my school to learn enough to be to able to take on the role of a Director position. In my conversation with my head of school, we realized this, and realized that I would probably need to look at moving onto a new school or other role if I wanted a greater leadership role.
I sent off an email to my best friends around the world and asked their advice. Most of them responded, and after doing some independent job search through the various international schools around the world, I ended up with eight different job offers to explore, all of which entailed more of a leadership role than I currently have.
The most exciting of these opportunities came through my friend Andrew Stillman who works for New Visions for Public Schools in NYC. Andrew connected me to Janet Price, Director of Instruction for New Visions, who decided (after reviewing my resume and interviewing me with a team of others) to offer me a job as a Math Formative Assessment Specialist for New Visions, which I accepted. I start at the beginning of July.
My core duties at New Visions will be:
- Collaborate with MASTER program staff and faculty at Hunter College to design and implement new curriculum and assessments for MASTER residents focused around the development of pedagogical content knowledge in math.
- Collaborate with New Visions and Hunter College faculty on in-field coaching and assessment of mentors and residents. Work with the team to set standards, review candidates’ progress, and iteratively adjust the program design
- Curate tasks that support embedded formative assessment strategies and develop teacher training for effective teacher use of these tasks to support learning through units of algebra and geometry in the common a2i curriculum
- Meet weekly with school-based teams in two New Visions high schools to, through a collaborative inquiry process:
- Implement a common-core aligned curriculum in algebra and geometry
- Facilitate looking at student work on formative assessment tools, including pre-unit, post-unit and day-to-day units to continuously improve curriculum , inform future lessons and identify areas for re-engaging students in content and processes
- Facilitate the development of teachers’ pedagogical content knowledge and instructional strategy “toolboxes” through:
- Assisting teachers in developing their listening, questioning and feedback skills and their ability to engineer effective classroom tasks
- Leading the math teacher teams in an inquiry process within and across schools to continuously use information gleaned from student work to improve instruction and meet student needs
- Meet regularly with math coaches to, through a collaborative inquiry process, reflect on the teachers’ progress as evidenced by student work and revise and plan new interventions and supports to move teachers forward.
- Assist in planning and facilitating of cross-school inquiry.
- Build and sustain relationships with schools and principals and support capacity building in participating schools.
- Engage in 5 hours per week in peer-to-peer learning and support on a team of Instructional Specialists and Residency Coaches
I’m looking forward to this role, while at the same time feeling sad about leaving my current role, my colleagues, my students, and Vancouver (a city I have grown to love, despite its rain). Change is always uncomfortable, although this is somewhat easier for my wife and I, given that this will be the fourth different country we’ve moved to together. Also, I started my teaching career in NYC, and lived for three years, so I am fairly familiar with the city.
I’m going to miss British Columbia, but I have no intention of severing my connection to it. This role in NYC will not last forever, and I hope to keep intact my ties here in British Columbia for when my wife and I decide that we miss Vancouver and our friends too much, and want to return home.
Those of you who have been following my work for a while will know that I am passionate about two main areas in education, mathematics education and the application of technology to education. This role will help me pursue my passion for improving mathematics education, and knowing me, I will almost certainly bring in technology to whatever role I do.
Mark writes:
"Here’s a really interesting experiment to try, if you have the opportunity. Visit an elementary school classroom. First, just watch the teacher interact with the students while they’re teaching. Don’t try to count interactions. Just watch. See if you think that any group of kids is getting more attention than any other. Most of the time, you probably will get a feeling that they’re paying roughly equal attention to the boys and the girls, or to the white students and the black students. Then, come back on a different day, and count the number of times that they call on boys versus calling on girls. I’ve done this, after having the idea suggested by a friend. The result was amazing. I really, honestly believed that the teacher was treating her students (the teacher I did this with was a woman) equally. But when I counted?She was calling on boys twice as often as girls."
I’d love to try this experiment out at my school, but I suspect I will not, and will instead ask my colleagues to try it out on themselves. Like Mark writes, this does not happen because the teachers are sexist, I’m sure they do not feel that they are at all sexist. These problems are systemic in our society, and you need someone on the outside looking in to have a chance at noticing them.
I believe that social media has the potential to facilitate a transformation in education in a way that no other communication tool before it has.
First, social media allows teachers to learn about ideas outside of their school or school district. Too often we are isolated within our classrooms, within our schools, and within our school districts, and we make assumptions about how certain educational practices should be done. When we see other schools doing things differently, it makes us wonder how we could change or improve our own practices. While other broadcast mediums (such as print and television) have this capability, social media allows us to both find out about an alternative practice and discuss the details of implementing this practice directly with whomever has created it.
Professional learning for teachers is also changing. Educators can now use social media to connect with ideas any time, any place. The #edchat discussion that happens weekly on Twitter is more similar to an Edcamp than it is to a traditional conference. An enormous percentage of what teachers learn comes from informal settings, and social media can extend the times and places where this informal learning can take place.
Just like their students, educators also need to feel like part of a community, and in some schools, they may be too different from their peers to form emotional attachments within their school. Social media allows these educators to find a peer group outside of their school with whom they can connect and form communities of care. Educators who feel like they are part of a community have greater morale, and are better able to cope with the stress their jobs entail.
I have also found that social media both exposes educators to the big ideas of education and the "what can I do on Monday" type of resource. It is important to have both – the first because the big ideas of education are what drive change, and the second because having resources available to use on a daily basis give you time to think about the big ideas.
Social media allows educators to, as a network, collaborate to solve problems that none of them could individually solve. I recently started a presentation on formative assessment. I seeded it with 20 examples of formative assessment, and then sent a link to my network of educators asking for more examples. The presentation is now up to 55 examples, and I could not have come up with all of those examples myself.
There are also some problems (or maybe they are more accurately named opportunities?) with social media.
I have seen many examples (and participated in many examples) of miscommunication that occurs because of the general terseness of the medium, and sometimes because of a fundamental disagreement about what the language being used means. I had a half-an-hour-long argument with another educator which only ended when I realized that she was using a completely different definition of learning than me. It is important to take the time to clarify language, and where necessary, link to less concise explanations of what we mean. This is one reason why I think that every educator who participates in social media should have some web-space available to which they can link when necessary.
Social media also favors people who are already well-connected. I am able to use social media as an especially effective means of collaboration because I have many educators in my network already. For people who are just getting started with Twitter, they may see it as more of a means to follow people who broadcast, rather than as much of a tool for connecting with and discussing educational ideas with other educators. As someone who is well-connected, I do my best to share some of the good projects and ideas I see from people within my network, so that my network can be at least in part a shared resource.
It is also well-known that people tend to repeat opinions that are popular more than opinions held by a minority. We naturally have a desire to be part of a group, and one consequence is that we can sometimes fall into the trap of groupthink. This phenomena also happens at the school level! It is therefore important that every network should contain some dissenters, some people who are willing to go against the crowd. We also need to think about our reasons for believing something to be true – do we really believe it, do we have evidence to support our belief, or are we just following the crowd?
Social media by itself will not change education – that responsibility lies with the people who use it, but change starts with desire, and social media can provide information which may lead to a desire to change.
My youngest son recently learned how to walk. He’s certainly not an expert by any means, but he can now toddle around for 10 to 12 steps at a time and not fall down. I noticed something strange about his walking though – he makes no effort to avoid obstacles in his way.
This is strange because I remember when he first learned to crawl a few months ago, he had the same problem, but he fairly quickly learned how to crawl around the obstacles on the floor and therefore avoid them.
There are two possibilities – he knows about the obstacles and chooses not to avoid them because he wants to focus on keeping his balance, or he doesn’t notice the obstacles at all. The first possibility seems unlikely to me since he normally hesitates in his walking when he seems uncertain and he does not even look at the obstacles at the floor. The second possibility seems more likely to me.
How is it that he can not notice obstacles while walking that he avoids easily while crawling? My theory is that there are three factors at play – first that his perspective is different while walking than while crawling and that he is not able to adjust his internal model of "looking for an avoiding obstacles" because of this. Another is that he does think that he should worry about obstacles while walking because it does not occur to him that the obstacles he successfully avoided while crawling could be obstacles while walking. Finally, walking probably takes my son quite a bit of cognitive effort. He has to spend his time thinking about balancing, and he is unable to think about all of the other things that could happen while he is doing this. Walking is not something for which he has automaticity. All of these reasons are results of a change in context between crawling and walking.
The space is the same, the obstacles are the same, but the context is different, and this makes the tasks different enough that even what seem like simple obstacles to us are currently challenging for my son.
We need to consider this issue for our students as well. How often do we see them unable to solve problems in an even slightly different context that they seemed completely capable of solving before?
Listen to the two songs linked below, and ask yourself, is this the message we should be sending our children?
After the horrible rape case in the now infamous town of Steubenville, I have been thinking about what could possibly have made this act seem justified by the boys who committed it, and although I do not think we can lay all of the blame on popular media, as it is a reflection of our culture, some of the blame must lie there.
I listened to this song, at the request of my son, and I was quickly horrified. It reminded me of a song I knew growing up, and how my uncritical mind had been deceived into liking this song, until a friend quietly pointed out (while I was singing the song) what the lyrics to the song meant.
I remember the moment that opened my mind, and started me thinking about music more critically.
My friend and I were walking from Koerner Library to the Student Union building on UBC campus, as part of our weekly Safewalk shift, and I started to hum, and then sing. As I got to "how easy it would be to show me how you feel", my friend interrupted me and asked, "Do you know what those lyrics mean?" I stopped singing, and said, "Uh…" I was slightly embarrassed. "That song is about pressuring girls to have sex with their boyfriend," she continued, "Are you sure you want to be singing it?" I paused, and ran through the lyrics in my head, and realized just how right she was.
I will admit that at the time I had more than my fair share of naïveté, but I believe that this is a common experience for many to not think very critically about the music to which they listen (or any media which they consume, for that matter).
I certainly know that young children, like my son, are especially unlikely to think critically about music. I wonder where my son learned of this song, and who introduced it to him and I wonder if they talked about the meaning of the song. I am especially worried that songs like this will influence his developing perspective on women, and his later relationship to them.
I would like my son, and all other boys, to grow up to be men of which we can be proud. Please, if you are exposing children to music or any other media, please, please think about what music to which you expose them, and ask yourself, if this child accepted the message of this music whole-heartedly, would this make them a better person?
- Do not feel absolutely certain of anything.
- Do not think it worth while to proceed by concealing evidence, for the evidence is sure to come to light.
- Never try to discourage thinking for you are sure to succeed.
- When you meet with opposition, even if it should be from your husband or your children, endeavour to overcome it by argument and not by authority, for a victory dependent upon authority is unreal and illusory.
- Have no respect for the authority of others, for there are always contrary authorities to be found.
- Do not use power to suppress opinions you think pernicious, for if you do the opinions will suppress you.
- Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.
- Find more pleasure in intelligent dissent that in passive agreement, for, if you value intelligence as you should, the former implies a deeper agreement than the latter.
- Be scrupulously truthful, even if the truth is inconvenient, for it is more inconvenient when you try to conceal it.
- Do not feel envious of the happiness of those who live in a fool’s paradise, for only a fool will think that it is happiness.
From Bertrand Russell’s "A Liberal Decalogue" via the Brain Pickings blog.
A few weeks ago I read someone (whom I cannot remember…) predict that Google Reader would soon disappear, and this person was right. Fortunately, I took the advice at the time and started looking into alternatives as Google just announced that Google Reader will be shut down in July.
First, I exported all of my RSS feeds from Google Reader. To do this, I needed to find the export option, which is hidden in the settings page.
Next, I had to find the export link. I noticed right away that this exporting process used to be a LOT easier. Sigh.
Clicking on this link took me to a sign-in page for Google’s Takeout service (which seems to be a utility for exporting any of your data out of Google). After a couple of failures, I finally managed to download an archive of my Google Reader information.
Clicking on Create Archive generated an archive of my RSS subscriptions, which I then had to download and extract (I recommend 7Zip for this). Inside this archive was the one file I wanted, the OPML file for my subscriptions. I knew I would need this file so I could import my subscriptions into another service.
Once I had a copy of my subscriptions in place, I looked into desktop alternatives, and found Thunderbird, Outlook, RSS Bandit, and RSS Owl. After installing RSS Bandit and successfully importing all of RSS feeds into it, I realized I did not want a desktop application because I would end up only ever being able to read my RSS feeds from that specific computer. I really appreciated the portability of Google Reader, and this was one feature I didn’t want to lose.
I had heard about other online RSS readers, but after Google Reader announced that they were shutting down, I decided that I would prefer to try hosting my own online RSS reader. I had a suspicion that some open source projects would exist, and with some minimal searching, I found three options: Tiny Tiny RSS, Selfoss, and RSS Lounge.
I tried to install Tiny Tiny RSS, but for some reason it failed, and so I quickly gave up on it (given I had other options). I then looked at some screen-shots of Selfoss and RSS Lounge, and decided I liked the look and feel of RSS Lounge better, so I installed it instead.
Installing RSS Lounge was relatively easy as web applications go. I had to extract the files from the archive I downloaded, and uploaded them to my site using Filezilla into a subfolder on this website. Next, I created a database for RSS Lounge using Cpanel, set the file permissions as per the installation instructions, and then navigated to the online installation page.
I entered in the information required for the database, and chose a username and password for the site. I also chose to keep the public access option unchecked, not because I am particularly concerned about other people knowing what I am reading, but because I want to keep extra traffic to my site to a minimum. Next, I logged into my new RSS reader, and found the OPML import option. I selected my subscriptions file and uploaded it to the site.
At first, I thought nothing had happened, until eventually the page refreshed and I saw a progress bar filling up. It took a very long time for all of the subscriptions to be updated, as each of the 729 blogs I am currently subscribed to had to have it’s RSS feed fetched, parsed, and "new" entries added to the site.
Once all of the RSS feeds were updated, I had over 4000 "unread" posts. Of course, I had actually already at least scanned all of these posts in Google Reader and marked them as read, so I started to mark them manually as read, 50 posts at a time. After 3 or 4 minutes of clicking, I gave up, and ran the following command in my database which updated all of my unread items as read. Whew!
I’ve now completely migrated away from Google Reader, and have access to my RSS feeds in an online program that I can control myself. The only reason I’ve used Google Reader since is to verify that my online reader is in fact fetching all of my feeds, which it apparently is.
